Do LED lights have a ballast? No.
An LED works on a rectified (basically constant) voltage. AC is sinusoidal. DC, is near a straight line. Thus, the light strips you bought, at some point, convert AC to DC.
LED's use very little current and just as low voltage, so either the bulbs/strips have a voltage converter in them, or the "power supply" that came with them is your power converter.
The power converter will die. The LED's should "last a lifetime" .... but not "forever".
If there was lightning strikes, or power surges in your home, this can cause premature failure. The power output of an LED strip should be regulated, but to save a few bucks the manufacturer might have gone cheap and just stepped the voltage down without regulation. Either way, the point here is "just because you paid a lot doesn't mean it is a quality product".
Can you replace the bad parts? .... It depends. There isn't going to be much to the lights so board level repair COULD have a fuse, but that would be added cost the manufacturer may not have put in. The other parts (electrolytic caps, bridge rectifier, MOV, resistors, regulator, transformer) are all possibilities but accessing them could be a challenge. If you are feeling adventurous, remove one and disassemble it, see what you have access to, post a few pictures and maybe we can tell you what to test out.
Should you go back to Fluorescent? My answer, No. Why? They have ballasts, they go bad, they cost $ to run, and now that we are a few years past your last purchase, you should be able to get replacements for less than your initial investment.